Inprojective geometry, ahomography is anisomorphism ofprojective spaces, induced by an isomorphism of thevector spaces from which the projective spaces derive.^[1] It is abijection that mapslines to lines, and thus acollineation. In general, some collineations are not homographies, but thefundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms includeprojectivity,projective transformation, andprojective collineation.

Historically, homographies (and projective spaces) have been introduced to studyperspective andprojections inEuclidean geometry, and the termhomography, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which extendedEuclidean andaffine spaces by the addition of new points calledpoints at infinity. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of avector space over a givenfield (the above definition is based on this version); this construction facilitates the definition ofprojective coordinates and allows using the tools oflinear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see alsosynthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".

For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative)field. EquivalentlyPappus's hexagon theorem andDesargues's theorem are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.

Historically, the concept of homography had been introduced to understand, explain and studyvisual perspective, and, specifically, the difference in appearance of two plane objects viewed from different points of view.

In three-dimensional Euclidean space, acentral projection from a pointO (the center) onto a planeP that does not containO is the mapping that sends a pointA to the intersection (if it exists) of the lineOA and the planeP. The projection is not defined if the pointA belongs to the plane passing throughO and parallel toP. The notion ofprojective space was originally introduced by extending the Euclidean space, that is, by addingpoints at infinity to it, in order to define the projection for every point exceptO.

Given another planeQ, which does not containO, therestriction toQ of the above projection is called aperspectivity.

With these definitions, a perspectivity is only apartial function, but it becomes abijection if extended to projective spaces. Therefore, this notion is normally defined for projective spaces. The notion is also easily generalized to projective spaces of any dimension, over anyfield, in the following way:

Given two projective spacesP andQ of dimensionn, aperspectivity is a bijection fromP toQ that may be obtained by embeddingP andQ in a projective spaceR of dimensionn + 1 and restricting toP a central projection ontoQ.

Iff is a perspectivity fromP toQ, andg a perspectivity fromQ toP, with a different center, theng ⋅f is a homography fromP to itself, which is called acentral collineation, when the dimension ofP is at least two. (See§ Central collineations below andPerspectivity § Perspective collineations.)

Originally, ahomography was defined as thecomposition of a finite number of perspectivities.^[2] It is a part of the fundamental theorem of projective geometry (see below) that this definition coincides with the more algebraic definition sketched in the introduction and detailed below.

Aprojective space P(V) of dimensionn over afieldK may be defined as the set of the lines through the origin in aK-vector spaceV of dimensionn + 1. If a basis ofV has been fixed, a point ofV may be represented by a point(x₀, ...,x_n) ofKⁿ⁺¹. A point of P(V), being a line inV, may thus be represented by the coordinates of any nonzero point of this line, which are thus calledhomogeneous coordinates of the projective point.

Given two projective spaces P(V) and P(W) of the same dimension, ahomography is a mapping from P(V) to P(W), which is induced by anisomorphism of vector spacesf :V →W. Such an isomorphism induces abijection from P(V) to P(W), because of the linearity off. Two such isomorphisms,f andg, define the same homographyif and only if there is a nonzero elementa ofK such thatg =af.

This may be written in terms of homogeneous coordinates in the following way: A homographyφ may be defined by anonsingular(n+1) × (n+1) matrix [a_i,j], called thematrix of the homography. This matrix is definedup to the multiplication by a nonzero element ofK. The homogeneous coordinates[x₀ : ... :x_n] of a point and the coordinates[y₀ : ... :y_n] of its image byφ are related by

When the projective spaces are defined by addingpoints at infinity toaffine spaces (projective completion) the preceding formulas become, in affine coordinates,

which generalizes the expression of the homographic function of the next section. This defines only apartial function between affine spaces, which is defined only outside thehyperplane where the denominator is zero.

The projective line over afieldK may be identified with the union ofK and a point, called the "point at infinity" and denoted by ∞ (seeProjective line). With this representation of the projective line, the homographies are the mappings

${\displaystyle z\mapsto \frac{az+b}{cz+d},\text{ where }ad-bc\ne 0,}$

which are calledhomographic functions orlinear fractional transformations.

In the case of thecomplex projective line, which can be identified with theRiemann sphere, the homographies are calledMöbius transformations.These correspond precisely with those bijections of the Riemann sphere that preserve orientation and are conformal.^[3]

In the study of collineations, the case of projective lines is special due to the small dimension. When the line is viewed as a projective space in isolation, anypermutation of the points of a projective line is a collineation,^[4] since every set of points are collinear. However, if the projective line is embedded in a higher-dimensional projective space, the geometric structure of that space can be used to impose a geometric structure on the line. Thus, in synthetic geometry, the homographies and the collineations of the projective line that are considered are those obtained by restrictions to the line of collineations and homographies of spaces of higher dimension. This means that the fundamental theorem of projective geometry (see below) remains valid in the one-dimensional setting. A homography of a projective line may also be properly defined by insisting that the mapping preservescross-ratios.^[5]

Aprojective frame orprojective basis of a projective space of dimensionn is an ordered set ofn + 2 points such that no hyperplane containsn + 1 of them. A projective frame is sometimes called asimplex,^[6] although asimplex in a space of dimensionn has at mostn + 1 vertices.

Projective spaces over a commutative fieldK are considered in this section, although most results may be generalized to projective spaces over adivision ring.

LetP(V) be a projective space of dimensionn, whereV is aK-vector space of dimensionn + 1, andp :V ∖ {0} →P(V) be the canonical projection that maps a nonzero vector to the vector line that contains it.

For every frame ofP(V), there exists a basise₀, ...,e_n ofV such that the frame is(p(e₀), ...,p(e_n),p(e₀ + ... +e_n)), and this basis is unique up to the multiplication of all its elements by the same nonzero element ofK. Conversely, ife₀, ...,e_n is a basis ofV, then(p(e₀), ...,p(e_n),p(e₀ + ... +e_n)) is a frame ofP(V)

It follows that, given two frames, there is exactly one homography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is theidentity map. This result is much more difficult insynthetic geometry (where projective spaces are defined through axioms). It is sometimes called thefirst fundamental theorem of projective geometry.^[7]

Every frame(p(e₀), ...,p(e_n),p(e₀ + ... +e_n)) allows to defineprojective coordinates, also known ashomogeneous coordinates: every point may be written asp(v); the projective coordinates ofp(v) on this frame are the coordinates ofv on the base(e₀, ...,e_n). It is not difficult to verify that changing thee_i andv, without changing the frame norp(v), results in multiplying the projective coordinates by the same nonzero element ofK.

The projective spaceP_n(K) =P(Kⁿ⁺¹) has acanonical frame consisting of the image byp of the canonical basis ofKⁿ⁺¹ (consisting of the elements having only one nonzero entry, which is equal to 1), and(1, 1, ..., 1). On this basis, the homogeneous coordinates ofp(v) are simply the entries (coefficients) of thetuplev. Given another projective spaceP(V) of the same dimension, and a frameF of it, there is one and only one homographyh mappingF onto the canonical frame ofP_n(K). The projective coordinates of a pointa on the frameF are the homogeneous coordinates ofh(a) on the canonical frame ofP_n(K).

In above sections, homographies have been defined through linear algebra. Insynthetic geometry, they are traditionally defined as the composition of one or several special homographies calledcentral collineations. It is a part of the fundamental theorem of projective geometry that the two definitions are equivalent.

In a projective space,P, of dimensionn ≥ 2, acollineation ofP is a bijection fromP ontoP that maps lines onto lines. Acentral collineation (traditionally these were calledperspectivities,^[8] but this term may be confusing, having another meaning; seePerspectivity) is a bijectionα fromP toP, such that there exists ahyperplaneH (called theaxis ofα), which is fixed pointwise byα (that is,α(X) =X for all pointsX inH) and a pointO (called thecenter ofα), which is fixed linewise byα (any line throughO is mapped to itself byα, but not necessarily pointwise).^[9] There are two types of central collineations.Elations are the central collineations in which the center is incident with the axis andhomologies are those in which the center is not incident with the axis. A central collineation is uniquely defined by its center, its axis, and the imageα(P) of any given pointP that differs from the centerO and does not belong to the axis. (The imageα(Q) of any other pointQ is the intersection of the line defined byO andQ and the line passing throughα(P) and the intersection with the axis of the line defined byP andQ.)

A central collineation is a homography defined by a(n+1) × (n+1) matrix that has aneigenspace of dimensionn. It is a homology, if the matrix has anothereigenvalue and is thereforediagonalizable. It is an elation, if all the eigenvalues are equal and the matrix is not diagonalizable.

The geometric view of a central collineation is easiest to see in a projective plane. Given a central collineationα, consider a line ℓ that does not pass through the centerO, and its image underα,ℓ′ =α(ℓ). SettingR = ℓ ∩ ℓ′, the axis ofα is some lineM throughR. The image of any pointA of ℓ underα is the intersection ofOA with ℓ′. The imageB′ of a pointB that does not belong to ℓ may be constructed in the following way: letS =AB ∩M, thenB′ =SA′ ∩OB.

The composition of two central collineations, while still a homography in general, is not a central collineation. In fact, every homography is the composition of a finite number of central collineations. In synthetic geometry, this property, which is a part of the fundamental theory of projective geometry is taken as the definition of homographies.^[10]

There are collineations besides the homographies. In particular, anyfield automorphismσ of a fieldF induces a collineation of every projective space overF by applyingσ to allhomogeneous coordinates (over a projective frame) of a point. These collineations are calledautomorphic collineations.

Thefundamental theorem of projective geometry consists of the three following theorems.

1. Given two projective frames of a projective spaceP, there is exactly one homography ofP that maps the first frame onto the second one.
2. If the dimension of a projective spaceP is at least two, every collineation ofP is the composition of an automorphic collineation and a homography. In particular, over the reals, every collineation of a projective space of dimension at least two is a homography.^[11]
3. Every homography is the composition of a finite number ofperspectivities. In particular, if the dimension of the implied projective space is at least two, every homography is the composition of a finite number of central collineations.

If projective spaces are defined by means of axioms (synthetic geometry), the third part is simply a definition. On the other hand, if projective spaces are defined by means oflinear algebra, the first part is an easy corollary of the definitions. Therefore, the proof of the first part in synthetic geometry, and the proof of the third part in terms of linear algebra both are fundamental steps of the proof of the equivalence of the two ways of defining projective spaces.

As every homography has aninverse mapping and thecomposition of two homographies is another, the homographies of a given projective space form agroup. For example, theMöbius group is the homography group of any complex projective line.

As all the projective spaces of the same dimension over the same field are isomorphic, the same is true for their homography groups. They are therefore considered as a single group acting on several spaces, and only the dimension and the field appear in the notation, not the specific projective space.

Homography groups also calledprojective linear groups are denotedPGL(n + 1,F) when acting on a projective space of dimensionn over a fieldF. Above definition of homographies shows thatPGL(n + 1,F) may be identified to thequotient groupGL(n + 1,F) /F^×I, whereGL(n + 1,F) is thegeneral linear group of theinvertible matrices, andF^×I is the group of the products by a nonzero element ofF of theidentity matrix of size(n + 1) × (n + 1).

WhenF is aGalois field GF(q) then the homography group is writtenPGL(n,q). For example,PGL(2, 7) acts on the eight points in the projective line over the finite field GF(7), whilePGL(2, 4), which is isomorphic to thealternating group A₅, is the homography group of the projective line with five points.^[12]

The homography groupPGL(n + 1,F) is a subgroup of thecollineation groupPΓL(n + 1,F) of the collineations of a projective space of dimensionn. When the points and lines of the projective space are viewed as ablock design, whose blocks are the sets of points contained in a line, it is common to call the collineation group theautomorphism group of the design.

The cross-ratio of four collinear points is an invariant under the homography that is fundamental for the study of the homographies of the lines.

Three distinct pointsa,b andc on a projective line over a fieldF form a projective frame of this line. There is therefore a unique homographyh of this line ontoF ∪ {∞} that mapsa to∞,b to 0, andc to 1. Given a fourth point on the same line, thecross-ratio of the four pointsa,b,c andd, denoted[a,b;c,d], is the elementh(d) ofF ∪ {∞}. In other words, ifd hashomogeneous coordinates[k : 1] over the projective frame(a,b,c), then[a,b;c,d] =k.^[13]

SupposeA is aring andU is itsgroup of units. Homographies act on a projective line overA, written P(A), consisting of pointsU[a, b] withprojective coordinates. The homographies on P(A) are described by matrix mappings

WhenA is acommutative ring, the homography may be written

${\displaystyle z\mapsto \frac{za+b}{zc+d},}$

but otherwise the linear fractional transformation is seen as an equivalence:

${\displaystyle U[za+b,zc+d]\sim U[(zc+d{)}^{-1}(za+b),1].}$

The homography group of the ring ofintegersZ ismodular groupPSL(2,Z). Ring homographies have been used inquaternion analysis, and withdual quaternions to facilitatescrew theory. Theconformal group of spacetime can be represented with homographies whereA is thecomposition algebra ofbiquaternions.^[14]

The homography isperiodic when the ring isZ/nZ (theintegers modulon) since thenArthur Cayley was interested in periodicity when he calculated iterates in 1879.^[15]In his review of a brute force approach to periodicity of homographies,H. S. M. Coxeter gave this analysis:

A real homography is involutory (of period 2) if and only ifa +d = 0. If it is periodic with periodn > 2, then it is elliptic, and no loss of generality occurs by assuming thatad −bc = 1. Since the characteristic roots are exp(±hπi/m), where(h,m) = 1, thetrace isa +d = 2 cos(hπ/m).^[16]

• W-curve

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